122. The ambiguity of his position will become clearer if we resort to his favourite ‘instances in gold.’ The proposition, ‘all gold is soluble in aqua regia,’ is certainly true, if such solubility is included in the complex idea which the word ‘gold’ stands for, and if such inclusion is all that the proposition purports to state. It is equally certain and equally trifling with the proposition, ‘a centaur is four-footed.’ But, in fact, as a proposition concerning substance, it purports to state more than this, viz. that a ‘body whose complex idea is made up of yellow, very weighty, ductile, fusible, and fixed,’ is always soluble in aqua regia. In other words, it states the invariable co-existence in a body of the complex idea, ‘solubility in aqua regia,’ with the group of ideas indicated by ‘gold.’ Thus understood--as instructive or synthetical--it has not the certainty which would belong to it if it were ‘trifling,’ or analytical, ‘since we can never, from the consideration of the ideas themselves, with certainty affirm’ their co-existence (Book IV. chap. vi. sec 9). If we see the solution actually going on, or can recall the sight of it by memory, we can affirm its co-existence with the ideas in question in that ‘bare instance;’ and thus, on the principle that ‘whatever ideas have once been united in nature may be so united again’ (Book IV. chap. iv. sec. 12), infer a capacity of co-existence between the ideas, but that is all. ‘Constant observation may assist our judgments in guessing’ an invariable actual co-existence (Book IV. chap. viii. sec. 9); but beyond guessing we cannot get. If our instructive proposition concerning co-existence is to be general it must remain problematical. It is otherwise with mathematical propositions. ‘If the three angles of a triangle were once equal to two right angles, it is certain that they always will be so;’ but only because such a proposition concerns merely ‘the habitudes and relations of ideas.’ ‘If the perception that the same ideas will eternally have the same habitudes and relations be not a sufficient ground of knowledge, there could be no knowledge of general propositions in mathematics; for no mathematical demonstration could be other than particular: and when a man had demonstrated any proposition concerning one triangle and circle, his knowledge would not reach beyond that particular diagram’ (Book IV. chap. i. sec. 9). Not the knowledge which is now supposed to be got by induction. Yet more than Locke was entitled to suppose it could give.